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# Cubic Semisymmetric Graph

A cubic semisymmetric graph is a graph that is both cubic (i.e., regular of degree 3) and semisymmetric (i.e., edge- but not vertex-transitive).

The four smallest cubic semisymmetric graphs are illustrated above. The smallest of these is Gray graph on 54 vertices, the next smallest is the Iofinova-Ivanov graph on 110 vertices (Iofinova and Ivanov 2002, Marušič et al. 2005), the third is the Ljubljana graph (Conder et al. 2002), and the fourth is the Tutte 12-cage.

Some cubic semisymmetric graphs are summarized in the following table.

 graph 54 Gray graph 110 110-Iofinova-Ivanov graph 112 Ljubljana graph 126 Tutte 12-cage 182 182-Iofinova-Ivanov graph 506 506-Iofinova-Ivanov graph 990 990-Iofinova-Ivanov graph

Cubic Graph, Iofinova-Ivanov Graphs, Ljubljana Graph, Semisymmetric Graph, Tutte 12-Cage

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## References

Bouwer, I. A. "On Edge But Not Vertex Transitive Regular Graphs." J. Combin. Th. Ser. B 12, 32-40, 1972.Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Ljubljana Graph." 2002. http://citeseer.ist.psu.edu/conder02ljubljana.html.Ivanov, A. V. "On Edge But Not Vertex Transitive Regular Graphs." In Combinatorial Design Theory (Ed. C. J. Colbourn and R. Mathon). Amsterdam, Netherlands: North-Holland, pp. 273-285, 1987.Iofinova, M. E. and Ivanov, A. A. "Bi-Primitive Cubic Graphs." In Investigations in the Algebraic Theory of Combinatorial Objects. pp. 123-134, 2002. (Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, pp. 137-152, 1985.)Marušič, D.; Pisanski, T.; and Wilson, S. "The Genus of the Gray Graph is 7." Europ. J. Combin. 26, 377-385, 2005.

## Referenced on Wolfram|Alpha

Cubic Semisymmetric Graph

## Cite this as:

Weisstein, Eric W. "Cubic Semisymmetric Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubicSemisymmetricGraph.html